Quantum_Mechanics_Flashcards_Part1_2

Question 1

What equation represents a simple sine wave in quantum mechanics?

Answer 1

Ψ(x, t) = A sin(kx − ωt)

Question 2

What is the phase velocity of a wave?

Answer 2

v_phase = ω / k

Question 3

What is the complex form of a wave used in quantum mechanics?

Answer 3

Ψ(x, t) = A exp[i(kx − ωt)]

Question 4

How is probability related to the wave function in quantum mechanics?

Answer 4

P(x, t) = |Ψ(x, t)|² = Ψ*Ψ

Question 5

What principle relates localization in space to uncertainty in momentum?

Answer 5

Heisenberg Uncertainty Principle: Δx Δp ≥ ℏ/2

Question 6

What are the assumptions used to derive the Schrödinger Equation?

Answer 6

1. E = ℏω, 2. p = ℏk, 3. Energy conservation E = T + V

Question 7

What is the time-dependent Schrödinger Equation?

Answer 7

iℏ ∂Ψ/∂t = [−ℏ²/(2m) ∂²/∂x² + V(x,t)]Ψ

Question 8

What is the time-independent Schrödinger Equation?

Answer 8

−ℏ²/(2m) d²ψ/dx² + V(x)ψ = Eψ

Question 9

What does the wavefunction Ψ(x, t) represent?

Answer 9

A complete description of a quantum state.

Question 10

What are the properties of the probability density function P(x, t)?

Answer 10

Real, non-negative, normalized: ∫|Ψ|² dx = 1

Question 11

How is the expectation value of position defined?

Answer 11

⟨x⟩ = ∫ Ψ* x Ψ dx

Question 12

What is the momentum operator in 1D?

Answer 12

p̂ = −iℏ ∂/∂x

Question 13

What is the expectation value of momentum?

Answer 13

⟨p⟩ = ∫ Ψ* (−iℏ ∂/∂x) Ψ dx

Question 14

What is the kinetic energy operator?

Answer 14

T̂ = −ℏ²/(2m) ∂²/∂x²

Question 15

What is the Hamiltonian operator?

Answer 15

Ĥ = T̂ + V̂ = −ℏ²/(2m) ∂²/∂x² + V(x)

Question 16

How is a step potential defined in quantum mechanics?

Answer 16

V(x) = 0 for x ≤ 0, V₀ for x > 0

Question 17

What are the boundary conditions at a potential step?

Answer 17

Continuity of ψ and dψ/dx at the boundary

Question 18

What is quantum tunnelling?

Answer 18

Penetration of a particle into a classically forbidden region where E < V

Question 19

What is the transmission probability through a potential step?

Answer 19

T = 4qk / (q + k)²

Question 20

What is the reflection probability?

Answer 20

R = ((k − q)/(k + q))²

Question 21

What are the boundary conditions in an infinite square well?

Answer 21

ψ = 0 at the walls, ψ must be continuous inside

Question 22

What are the energy eigenvalues for an infinite square well?

Answer 22

Eₙ = n²π²ℏ²/(2mL²), n = 1,2,3,…

Question 23

What does the quantum number n represent?

Answer 23

It labels the allowed energy levels of a bound system

Question 24

What is zero-point energy?

Answer 24

The lowest possible energy of a quantum system, not zero due to HUP

Question 25

What is the potential energy in the quantum harmonic oscillator?

Answer 25

V(x) = ½ mω²x²

Question 26

What are the energy eigenvalues of the quantum harmonic oscillator?

Answer 26

Eₙ = (n + ½)ℏω

Question 27

What type of functions are the wavefunctions for the quantum harmonic oscillator?

Answer 27

ψₙ(x) = Hₙ(u) e^(−u²/2), where Hₙ are Hermite polynomials

Question 28

What does the time-dependent wavefunction Ψ(x,t) describe?

Answer 28

The full quantum state, including time evolution.

Question 29

What is a stationary state?

Answer 29

A state where the probability density |Ψ|² is time-independent.

Question 30

What causes oscillations in probability density for a superposition state?

Answer 30

Interference between energy eigenstates with different eigenvalues.

Question 31

What is Postulate 3 in quantum mechanics?

Answer 31

Any wavefunction can be expressed as a linear combination of eigenstates of a complete basis.

Question 32

What is the Copenhagen interpretation of measurement?

Answer 32

A measurement collapses the wavefunction to a corresponding eigenstate.

Question 33

What are Hermitian operators?

Answer 33

Operators with real eigenvalues; they represent observable quantities.

Question 34

What is orthogonality of eigenfunctions?

Answer 34

Eigenfunctions of a Hermitian operator are orthogonal: ∫ψₙ*ψₘ dx = δₙₘ

Question 35

How is the expectation value of an operator A in a superposition state calculated?

Answer 35

⟨A⟩ = ∑|cₙ|²aₙ, where aₙ are eigenvalues and cₙ are coefficients.

Question 36

What does it mean for two operators to commute?

Answer 36

Their commutator is zero: [A, B] = 0, so they can share eigenfunctions.

Question 37

Can non-commuting operators be simultaneously measured?

Answer 37

No, their measurements affect one another.

Question 38

What is the commutator of position and momentum?

Answer 38

[x, p̂] = iℏ

Question 39

What is the Heisenberg uncertainty principle?

Answer 39

Δx Δp ≥ ℏ/2 for position and momentum.

Question 40

Why do non-commuting operators lead to uncertainty?

Answer 40

Because their eigenfunctions are not shared.

Question 41

How does Fourier analysis relate to uncertainty?

Answer 41

A narrow wavepacket in position has a wide distribution in momentum.

Question 42

What is the 3D time-independent Schrödinger equation?

Answer 42

−ℏ²/(2m) ∇²ψ + V(r)ψ = Eψ

Question 43

What is the Laplacian in Cartesian coordinates?

Answer 43

∇² = ∂²/∂x² + ∂²/∂y² + ∂²/∂z²

Question 44

What is a separable solution for a 3D box?

Answer 44

ψ(x,y,z) = X(x)Y(y)Z(z), where each satisfies a 1D TISE.

Question 45

What is the energy for a particle in a 3D box?

Answer 45

E = (ℏ²π²/2m)(n₁²/a² + n₂²/b² + n₃²/c²)

Question 46

What is degeneracy in a 3D box?

Answer 46

Different quantum number combinations result in the same energy.

Question 47

What is the classical expression for angular momentum?

Answer 47

L = r × p

Question 48

What are the components of the angular momentum operator?

Answer 48

L̂ₓ = −iℏ(y∂/∂z − z∂/∂y), etc.

Question 49

Do angular momentum components commute?

Answer 49

No: [L̂ₓ, L̂ᵧ] = iℏL̂𝓏 and cyclic permutations.

Question 50

What is the operator for total angular momentum squared?

Answer 50

L̂² = L̂ₓ² + L̂ᵧ² + L̂𝓏²

Question 51

Which components of angular momentum can be known simultaneously?

Answer 51

L² and one component (usually L̂𝓏)

Question 52

What is the Coulomb potential for hydrogen?

Answer 52

V(r) = −Ze²/(4πε₀r)

Question 53

What are the three quantum numbers for hydrogen?

Answer 53

n (principal), l (orbital), m (magnetic)

Question 54

What is the energy of the nth level in hydrogen?

Answer 54

Eₙ = −13.6 Z² / n² eV

Question 55

What are spherical harmonics?

Answer 55

Yₗᵐ(θ, φ) = N Pₗᵐ(cosθ) e^{imφ}, eigenfunctions of angular momentum operators.

Question 56

What determines degeneracy in hydrogen?

Answer 56

Energy depends only on n; multiple (l, m) values share the same energy.